constant Cross product

Question

If vectors $\vec A,\vec B$ pass through points $P,Q$ and dot product

$$ \dfrac{\vec A\cdot \vec B}{|A||B|}=0, $$

then the locus of intersection points of $A,B$ is a sphere with diameter $PQ$

and if cross product

$$ \dfrac{|\vec A \times \vec B|}{|A||B|}=1, $$

then is the locus of intersection points of $A,B$ also a sphere with diameter $PQ?$

If not, what is the locus?

Answer 1

It is very well known that $$\dfrac{|\vec A \times \vec B|}{|A||B|}=|\sin\theta|$$where $\theta$ is the angle between $\vec {A}$ and $\vec{B}$. Also $|\sin\theta|=1$ is exactly equivalent to $\cos\theta=0$ i.e. they have the same roots. So what you argued is correct znd the locus is still the same sphere.

Answer 2

Using the triple product expansion identity $\,(a \times b) \cdot (c \times d) = (a \cdot c)(b \cdot d) - (a \cdot d)(b \cdot c)\,$:

$$ |A \times B|^2 = (A \times B) \cdot (A \times B) = |A|^2\,|B|^2 - (A \cdot B)^2 $$

It follows that $A \cdot B = 0 \iff |A \times B| = |A|\,|B|\,$.